By Ph. Boulanger, M. Hayes (auth.)
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Extra info for Bivectors and Waves in Mechanics and Optics
C= 1. 2) Using an orthonormal triad i, j, k of real unit vectors, the bivectors A, B, C may be written as A = A 1 i + A 2 j + A 3 k, B= B 1i + B 2 j + B 3k, C= C1i + Czj + C3k. 3) are as simple as possible. 5) for some real A. Now let k be defined by k = i x j. 8) for some complex B 0 , C0 . Now, B·B= C·C= 1 gives B~ + s; = 1, c~ + c; = 1. 10) C 0 =sin 'P', C 3 =cos 'P'. j)+sin'Pk}. 14c), we have (Ax B)·C= 1, and when the '-' sign is chosen, we have (A X B)·C= -1. 11 Reciprocal triads Let A, B, C be a basis (in general not orthonormal) for the hi vectors.
C. 3, it follows that C must not be isotropic, so that, without loss of generality, we take C· C = 1. Now, either Q possesses a nonisotropic eigenbivector corresponding to A, or all eigenbivectors corresponding to A. are isotropic. 8) B·B=tO. 1, we know that B is orthogonal to C: B· C = 0. Without loss of generality, we take B· B = 1, and let A = B x C. The bivectors A, B, C form an orthonormal triad. Let T be the matrix whose columns are A, B, C: T=(A IBIC). 9) Then it is easily seen that TTT= 1, TTQT= [AT;A 0 ~ ~].
Solve X x A +aX= B (a =f. 0). (a) When A· A + a2 =f. 0, show that the solution is unique. (b) When A· A + a2 = 0, show that there are no solutions unless the compatibility condition A x [aB +A x B] = 0 is satisfied. (c) When A· A + a 2 = 0 and when B is such that the compatibility condition is satisfied, show that one solution may be obtained as a linear combination of A and B. Then find all the solutions. 3. Solve the pair of equations X x A = B, x- C =a (with A· B = 0). 10 Orthonormal bases. Canonical form Let A, B, C be an orthonormal basis of bivectors: A·B= B·C= C.
Bivectors and Waves in Mechanics and Optics by Ph. Boulanger, M. Hayes (auth.)